Regression Standard Standard- T-Statistic Reject Power Independent Coefficient Error ized to Test Prob H0 at of Test Variable b(i) Sb(i) Coefficient H0: β(i)=0 Level 5%? at 5%
Intercept 85.24038 23.695135 0.0000 3.597 0.0058 Yes 0.8915 Test1 -1.93357 1.029096 -3.0524 -1.879 0.0930 No 0.3896 Test2 -1.65988 0.872896 -2.9224 -1.902 0.0897 No 0.3974 Test3 0.10495 0.219902 0.1404 0.477 0.6445 No 0.0713 Test4 3.77838 1.834497 4.7853 2.060 0.0695 No 0.4522 Test5 -0.04058 0.201221 -0.0595 -0.202 0.8447 No 0.0538
This section reports the values and significance tests of the regression coefficients. Before using this report, check that the assumptions are reasonable. For instance, collinearity can cause the t-tests to give false results and the regression coefficients to be of the wrong magnitude or sign.
Independent Variable
The names of the independent variables are listed here. The intercept is the value of the Y intercept.
Note that the name may become very long, especially for interaction terms. These long names may misalign the report. You can force the rest of the items to be printed on the next line by using the Skip Line After option in the Format tab. This should create a better looking report when the names are extra long.
Regression Coefficient
The regression coefficients are the least squares estimates of the parameters. The value indicates how much change in Y occurs for a one-unit change in that particular X when the remaining X’s are held constant. These coefficients are often called partial-regression coefficients since the effect of the other X’s is removed. These coefficients are the values of
b b
0, ,
1,b
p.Standard Error
The standard error of the regression coefficient, sbj, is the standard deviation of the estimate. It is used in hypothesis tests or confidence limits.
Standardized Coefficient
Standardized regression coefficients are the coefficients that would be obtained if you standardized the independent variables and the dependent variable. Here standardizing is defined as subtracting the mean and dividing by the standard deviation of a variable. A regression analysis on these standardized variables would yield these standardized coefficients.
When the independent variables have vastly different scales of measurement, this value provides a way of making comparisons among variables. The formula for the standardized regression coefficient is:
j, std j X Y
b
b
s
s
j=
wheres
Y ands
Xjare the standard deviations for the dependent variable and the j
th
independent variable.
T-Value to test Ho: B(i)=0
This is the t-test value for testing the hypothesis that
β
j=0
versus the alternative thatβ
j≠0
after removing the influence of all other X’s. This t-value has n-p-1 degrees of freedom.To test for a value other than zero, use the formula below. There is an easier way to test hypothesized values using confidence limits. See the discussion below under Confidence Limits. The formula for the t-test is
j j j * b
t
b
s
j=
−β
Prob LevelThis is the p-value for the significance test of the regression coefficient. The p-value is the probability that this t- statistic will take on a value at least as extreme as the actually observed value, assuming that the null hypothesis is true (i.e., the regression estimate is equal to zero). If the p-value is less than alpha, say 0.05, the null hypothesis of equality is rejected. This p-value is for a two-tail test.
Reject H0 at 5%?
This is the conclusion reached about the null hypothesis. It will be either reject H0 at the 5% level of significance or not.
Power (5%)
Power is the probability of rejecting the null hypothesis that
β
j=0
whenβ
j =β
j*≠0. The power is calculated for the case whenβ
j bj* =
,
σ
2=s
2, and alpha is as specified in the Alpha of C.I.’s and Tests option.High power is desirable. High power means that there is a high probability of rejecting the null hypothesis that the regression coefficient is zero when this is false. This is a critical measure of sensitivity in hypothesis testing. This estimate of power is based upon the assumption that the residuals are normally distributed.
Estimated Model
This is the least squares regression line presented in double precision. Besides showing the regression model in long form, it may be used as a transformation by copying and pasting it into the Transformation portion of the spreadsheet.
Note that a transformation must be less than 255 characters. Since these formulas are often greater than 255 characters in length, you must use the FILE(filename) transformation. To do so, copy the formula to a text file using Notepad, Windows Write, or Word to receive the model text. Be sure to save the file as an unformatted text (ASCII) file. The transformation is FILE(filename) where filename is the name of the text file, including directory information. When the transformation is executed, it will load the file and use the transformation stored there.